Subgroups — C₂₃×#3154 #15670773

Order 2944.

564 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#536	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#535	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#497	C₂	1	2	yes	maximal, complemented
8	C₂₃×#496	C₂	1	2	yes	maximal, complemented
9	C₂₃×#495	C₂	1	2	yes	maximal, complemented
10	C₂₃×#492	C₂	1	2	yes	maximal, complemented
11	C₂₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#137	—	2	4	no	—
17	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
18	C₂₃×#135	—	2	4	no	—
19	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
20	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	complemented
21	C₂×SD₁₆×C₂₃	—	2	4	no	—
22	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
23	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
24	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
25	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
26	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
27	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
28	C₂×D₈×C₂₃	C₂²	1	2	yes	—
29	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
30	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
33	C₂₃×#127	C₂²	1	2	yes	—
34	C₂₃×#124	C₂²	1	2	yes	—
35	C₂₃×#124	C₂²	1	2	yes	complemented
36	C₂₃×#123	—	2	2	no	—
37	C₂₃×C₂²⋊Q₈	—	2	2	no	—
38	C₂₃×#121	—	2	2	no	—
39	C₂₃×#120	—	2	2	no	—
40	C₂₃×#117	C₂²	1	1	yes	characteristic
41	C₂₃×#97	C₂²	1	2	yes	complemented
42	C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
43	C₂⁴×C₂₃	—	2	1	no	—
44	#41×C₂₃	—	2	2	no	—
45	#41×C₂₃	—	2	2	no	—
46	#41×C₂₃	—	2	2	no	—
47	#41×C₂₃	—	2	2	no	—
48	C₂×Q₈×C₂₃	—	2	1	no	—
49	C₂×Q₈×C₂₃	—	2	2	no	—
50	C₂×Q₈×C₂₃	—	2	2	no	—
51	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
53	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×D₄×C₂₃	—	2	1	no	—
55	C₂×D₄×C₂₃	—	2	1	no	—
56	C₂×D₄×C₂₃	—	2	2	no	—
57	C₂×D₄×C₂₃	—	2	2	no	—
58	C₂×D₄×C₂₃	—	2	1	no	—
59	C₂×D₄×C₂₃	—	2	4	no	—
60	C₂×D₄×C₂₃	C₂³	1	2	yes	—
61	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	—	2	1	no	—
64	C₂²×C₄×C₂₃	—	2	1	no	—
65	C₂²×C₄×C₂₃	—	2	1	no	—
66	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
67	Q₁₆×C₂₃	—	2	2	no	—
68	Q₁₆×C₂₃	—	2	4	no	—
69	SD₁₆×C₂₃	—	2	4	no	—
70	SD₁₆×C₂₃	—	2	4	no	—
71	SD₁₆×C₂₃	—	2	4	no	—
72	SD₁₆×C₂₃	—	2	2	no	—
73	SD₁₆×C₂₃	—	2	2	no	—
74	SD₁₆×C₂₃	—	2	4	no	—
75	D₈×C₂₃	—	2	4	no	—
76	D₈×C₂₃	—	2	2	no	—
77	M₁₆×C₂₃	D₄	1	4	yes	complemented
78	C₂×C₈×C₂₃	D₄	1	4	yes	complemented
79	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₂₃	C₂³	1	2	yes	—
82	C₄⋊C₄×C₂₃	—	2	4	no	—
83	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
84	C₂²⋊C₄×C₂₃	—	2	4	no	—
85	C₂²⋊C₄×C₂₃	—	2	4	no	—
86	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
87	C₄²×C₂₃	C₂³	1	2	yes	—
88	C₂³×C₂₃	—	2	1	no	—
89	C₂³×C₂₃	—	2	2	no	—
90	C₂³×C₂₃	—	2	1	no	—
91	C₂³×C₂₃	—	4	2	no	—
92	C₂³×C₂₃	—	8	2	no	—
93	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
94	Q₈×C₂₃	—	2	2	no	—
95	Q₈×C₂₃	—	2	1	no	—
96	Q₈×C₂₃	—	2	2	no	—
97	Q₈×C₂₃	—	2	1	no	—
98	Q₈×C₂₃	—	2	1	no	—
99	Q₈×C₂₃	—	2	1	no	—
100	Q₈×C₂₃	—	4	2	no	—
101	D₄×C₂₃	—	2	1	no	—
102	D₄×C₂₃	—	2	2	no	—
103	D₄×C₂₃	—	2	1	no	—
104	D₄×C₂₃	—	2	1	no	—
105	D₄×C₂₃	—	2	2	no	—
106	D₄×C₂₃	—	2	1	no	—
107	D₄×C₂₃	—	4	2	no	—
108	D₄×C₂₃	—	4	2	no	—
109	D₄×C₂₃	—	4	2	no	—
110	D₄×C₂₃	—	8	4	no	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	4	2	no	—
115	C₂×C₄×C₂₃	—	4	2	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
118	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
120	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
122	C₈×C₂₃	C₂×D₄	1	2	yes	—
123	C₈×C₂₃	C₂×D₄	1	2	yes	—
124	C₈×C₂₃	C₂×D₄	1	4	yes	—
125	#3154	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₂₃	—	2	2	no	—
127	C₂²×C₂₃	—	4	1	no	—
128	C₂²×C₂₃	—	4	1	no	—
129	C₂²×C₂₃	—	4	2	no	—
130	C₂²×C₂₃	—	4	2	no	—
131	C₂²×C₂₃	—	4	4	no	—
132	C₂²×C₂₃	—	4	2	no	—
133	C₂²×C₂₃	—	8	4	no	—
134	C₂²×C₂₃	#127	1	2	yes	—
135	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₂₃	—	2	4	no	—
137	C₄×C₂₃	—	4	1	no	—
138	C₄×C₂₃	—	4	1	no	—
139	C₄×C₂₃	—	4	2	no	—
140	C₄×C₂₃	#127	1	2	yes	—
141	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×#137	C₂×C₂₃	1	1	yes	characteristic, complemented
144	C₂×AGL(1,ℤ/8)	C₂×C₂₃	1	1	yes	characteristic, complemented
145	C₂×#135	C₂×C₂₃	1	1	yes	characteristic, complemented
146	C₂²×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
147	#536	C₂×C₂₃	1	1	yes	characteristic, complemented
148	#535	C₂×C₂₃	1	1	yes	characteristic, complemented
149	#497	C₂×C₂₃	1	2	yes	complemented
150	#496	C₂×C₂₃	1	2	yes	complemented
151	#495	C₂×C₂₃	1	2	yes	complemented
152	#492	C₂×C₂₃	1	2	yes	complemented
153	#405	C₂×C₂₃	1	1	yes	characteristic, complemented
154	C₂×C₂₃	—	2	2	no	—
155	C₂×C₂₃	—	8	1	no	—
156	C₂×C₂₃	—	8	1	no	—
157	C₂×C₂₃	—	8	2	no	—
158	C₂×C₂₃	C₂×#127	1	1	yes	characteristic
159	C₂×C₂₃	#577	1	2	yes	—
160	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
161	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
162	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
163	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
164	#137	—	2	4	no	—
165	AGL(1,ℤ/8)	—	2	4	no	—
166	#135	—	2	4	no	—
167	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
168	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	complemented
169	C₂×SD₁₆	—	2	4	no	—
170	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
171	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
172	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
173	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
174	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
175	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
176	C₂×D₈	C₂²×C₂₃	1	2	yes	—
177	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
178	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
179	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
180	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
181	#127	C₂²×C₂₃	1	2	yes	—
182	#124	C₂²×C₂₃	1	2	yes	—
183	#124	C₂²×C₂₃	1	2	yes	complemented
184	#123	—	2	2	no	—
185	C₂²⋊Q₈	—	2	2	no	—
186	#121	—	2	2	no	—
187	#120	—	2	2	no	—
188	#117	C₂²×C₂₃	1	1	yes	characteristic
189	#97	C₂²×C₂₃	1	2	yes	complemented
190	C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
191	C₂₃	#3154	1	1	yes	characteristic, direct product factor, Sylow
192	C₂⁴	—	2	1	no	—
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	#41	—	2	2	no	—
197	C₂×Q₈	—	2	1	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
201	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
202	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	C₂³×C₂₃	1	2	yes	—
210	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
211	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
216	Q₁₆	—	2	2	no	—
217	Q₁₆	—	2	4	no	—
218	SD₁₆	—	2	4	no	—
219	SD₁₆	—	2	4	no	—
220	SD₁₆	—	2	4	no	—
221	SD₁₆	—	2	2	no	—
222	SD₁₆	—	2	2	no	—
223	SD₁₆	—	2	4	no	—
224	D₈	—	2	4	no	—
225	D₈	—	2	2	no	—
226	M₁₆	D₄×C₂₃	1	4	yes	complemented
227	C₂×C₈	D₄×C₂₃	1	4	yes	complemented
228	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
229	C₂×C₈	C₂³×C₂₃	1	2	yes	—
230	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
236	C₄²	C₂³×C₂₃	1	2	yes	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	4	2	no	—
241	C₂³	—	8	2	no	—
242	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
243	Q₈	—	2	2	no	—
244	Q₈	—	2	1	no	—
245	Q₈	—	2	2	no	—
246	Q₈	—	2	1	no	—
247	Q₈	—	2	1	no	—
248	Q₈	—	2	1	no	—
249	Q₈	—	4	2	no	—
250	D₄	—	2	1	no	—
251	D₄	—	2	2	no	—
252	D₄	—	2	1	no	—
253	D₄	—	2	1	no	—
254	D₄	—	2	2	no	—
255	D₄	—	2	1	no	—
256	D₄	—	4	2	no	—
257	D₄	—	4	2	no	—
258	D₄	—	4	2	no	—
259	D₄	—	8	4	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
267	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
269	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
270	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
271	C₈	C₂×D₄×C₂₃	1	2	yes	—
272	C₈	C₂×D₄×C₂₃	1	2	yes	—
273	C₈	C₂×D₄×C₂₃	1	4	yes	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	8	4	no	—
282	C₂²	C₂₃×#127	1	2	yes	—
283	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	2	no	—
288	C₄	C₂₃×#127	1	2	yes	—
289	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
290	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	—	8	2	no	—
293	C₂	—	8	1	no	—
294	C₂	—	8	1	no	—
295	C₂	C₂×C₂₃×#127	1	1	yes	characteristic
296	C₂	C₂₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#117 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₂₃×#137	C₂	semidirect
2. C₂×C₂₃×AGL(1,ℤ/8)	C₂	semidirect
3. C₂×C₂₃×#135	C₂	semidirect
4. C₂²×SD₁₆×C₂₃	C₂	semidirect
5. C₂₃×#536	C₂	semidirect
6. C₂₃×#535	C₂	semidirect
7. C₂₃×#497	C₂	semidirect
8. C₂₃×#496	C₂	semidirect
9. C₂₃×#495	C₂	semidirect
10. C₂₃×#492	C₂	semidirect
11. C₂₃×#405	C₂	semidirect
19. C₂×Q₁₆×C₂₃	C₂²	semidirect
20. C₂×Q₁₆×C₂₃	C₂²	semidirect
25. C₂×SD₁₆×C₂₃	C₂²	semidirect
26. C₂×SD₁₆×C₂₃	C₂²	semidirect
27. C₂×SD₁₆×C₂₃	C₂²	semidirect
29. C₂×D₈×C₂₃	C₂²	semidirect
32. C₂₃×C₄⋊Q₈	C₂²	semidirect
35. C₂₃×#124	C₂²	semidirect
41. C₂₃×#97	C₂²	semidirect
42. C₄×C₈×C₂₃	C₂²	semidirect
77. M₁₆×C₂₃	D₄	semidirect
78. C₂×C₈×C₂₃	D₄	semidirect
125. #3154	C₂₃	direct (N × H)
143. C₂×#137	C₂×C₂₃	semidirect
144. C₂×AGL(1,ℤ/8)	C₂×C₂₃	semidirect
145. C₂×#135	C₂×C₂₃	semidirect
146. C₂²×SD₁₆	C₂×C₂₃	semidirect
147. #536	C₂×C₂₃	semidirect
148. #535	C₂×C₂₃	semidirect
149. #497	C₂×C₂₃	semidirect
150. #496	C₂×C₂₃	semidirect
151. #495	C₂×C₂₃	semidirect
152. #492	C₂×C₂₃	semidirect
153. #405	C₂×C₂₃	semidirect
167. C₂×Q₁₆	C₂²×C₂₃	semidirect
168. C₂×Q₁₆	C₂²×C₂₃	semidirect
173. C₂×SD₁₆	C₂²×C₂₃	semidirect
174. C₂×SD₁₆	C₂²×C₂₃	semidirect
175. C₂×SD₁₆	C₂²×C₂₃	semidirect
177. C₂×D₈	C₂²×C₂₃	semidirect
180. C₄⋊Q₈	C₂²×C₂₃	semidirect
183. #124	C₂²×C₂₃	semidirect
189. #97	C₂²×C₂₃	semidirect
190. C₄×C₈	C₂²×C₂₃	semidirect
191. C₂₃	#3154	direct (N × H)
226. M₁₆	D₄×C₂₃	semidirect
227. C₂×C₈	D₄×C₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#137	2
C₂×C₂₃×AGL(1,ℤ/8)	2
C₂×C₂₃×#135	2
C₂²×SD₁₆×C₂₃	2
C₂₃×#536	2
C₂₃×#535	2
C₂₃×#497	2
C₂₃×#496	2
C₂₃×#495	2
C₂₃×#492	2
C₂₃×#405	2
#3154	23

Subgroups — C23×#3154 #15670773

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3154 #15670773